Possibly because, before calculators and Excel made number work easier, we had to add up massive columns of figures, especially for bookkeeping and accounting. You started at the bottom and worked your way up
(and often lost count and had to start again) first the units, then carry over and do the tens etc etc until you finally got the correct total.
Then used for agreeing to someone's answer/suggestion to solve a general problem so "adds up (to correct answer)"
Must admit I loved Excel and don't miss doing maths the hard way!!
Work "out" (a problem)
Write "out" (an answer)
= possibly originated with maths - doing a formula/logic step by step until you get to the end
Now you've got me looking them all up!